Remark In order to get a \(\mathbb {C}P^1\) -reduction of the stable tangent bundle, we need to replace the inclusion \(i \colon \mathbb {C}P^1 \hookrightarrow \mathbb {C}P^{\infty }\)  in the pullback squares above by \(-i\) .
Replacing the basepoint inclusion \(* \rightarrow \mathbb {C}P^{\infty }\)  by the fibration \(S^{\infty } \rightarrow \mathbb {C}P^{\infty }\)  we obtain the definition given in [1]}:
\({Y [r] [d] & S^{\infty } [d] \\\mathit {BU}\times \mathbb {C}P^1 [r]^-{\det +i} & \mathbb {C}P^{\infty }}\) 

